Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre
Source: Originally published on arXiv quant-ph on July 20, 2026.
arXiv:2607.15326v1 Announce Type: new Abstract: Li and Boyle showed that the Penrose tiling defines a quantum error-correcting code: superpositions of tilings over isometry orbits protect quantum information against erasure of any bounded region. We extend the construction to the aperiodic monotiles discovered by Smith, Myers, Kaplan and Goodman-Strauss. For the Hat, we prove strong local indistinguishability for all Hat tilings, and we prove local recoverability unconditionally for all nonsingular Hat tilings via the torus parametrization of the underlying cut-and-project scheme. The remaining singular case reduces to one sharply posed geometric question -- can a region that is a union of hats be retiled a second way? -- which we verify computationally has no counterexample up to a substa...
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